The radiative heating rate is computed as the divergence of net radiation fluxes F so that. rad

Transcription

1 Chapter 2 Radiation Table of contents 2.1 Radiative heating 2.2 Longwave radiation The pre-cycle Cy22r3 scheme Vertical integration Spectral integration The incorporation of the effects of clouds The Rapid Radiation Transfer Model (RRTM) 2.3 Shortwave Radiation Spectral integration Vertical integration Multiple reflections between layers Cloud shortwave optical properties 2.4 Horizontal interpolation Temporal interpolation Spatial interpolation 2.5 Input to the radiation scheme Model variables Clouds Aerosols Carbon dioxide, ozone and trace gases Ground albedo and emissivity Solar zenith angle 2.6 The radiation code Set-up routines Main routines Specialized routines Heating rate computation Appendix A. List of symbols 2.1 RADIATIVE HEATING The radiative heating rate is computed as the divergence of net radiation fluxes F so that ( ) T = g F t c p p where c p is the specific heat at constant pressure of moist air rad (2.1) c p = c pdry {1 + (c pvap c pvap )q/c pdry } and c pdry and c pvap are the specific heats at constant pressure of dry air and water vapour, respectively. Sections 2.2 and 2.3 describe the computation of the longwave and shortwave radiative fluxes respectively. The solution of the radiative transfer equation to obtain the fluxes is unfortunately very expensive, and we IFS Documentation Cy31r1 7

2 Chapter 2: Radiation cannot afford to do it more than every 3 hours at every fourth grid point. The interpolation scheme used for obtaining the radiative fluxes at every grid point and every time step for the relevant instantaneous temperature profile and solar zenith angle is described in Section 2.4. A description of the inputs, in particular the climatologically defined quantities of radiative importance is given in Section 2.5. Finally, an alphabetical list of the subroutines of the radiation scheme is given in Section LONGWAVE RADIATION Since cycle Cy22r3, two longwave radiation schemes are available in the ECMWF model, the pre-cycle Cy22r3 by Morcrette (1991), and the current longwave radiation transfer scheme, the Rapid Radiation Transfer Model (RRTM). The rate of atmospheric cooling by emission-absorption of longwave radiation is T t = g F LW c p p where F LW is the net longwave radiation flux (the subscript LW is omitted in the remainder of this section). Assuming a non-scattering atmosphere in local thermodynamic equilibrium, F is given by 1 [ { 0 }] F = µ dµ dv L v (p surf, µ)t v (p surf, p, µ) + L v (p, µ) dt v p =p surf 1 0 where L v (p, µ) is the monochromatic radiance at wavenumber v at level p, propagating in a direction θ (the angle that this direction makes with the vertical), where µ = cos θ and t v (p, p ; r) is the monochromatic transmission through a layer whose limits are at p and p seen under the same angle θ, with r = sec θ. The subscript surf refers to the earth s surface. Subsections to describe the pre-cycle Cy22r3 scheme, and Subsections describes the RRTM scheme in Cy22r The pre-cycle Cy22r3 scheme After separating the upward and downward components (indicated by superscripts + and, respectively), and integrating by parts, we obtain the radiation transfer equation as it is actually estimated in the longwave part of the radiation code F v + (p) = [B v (T surf ) B v (T 0+ )]t v (p surf, p; r) + B v (T (p)) + 0 t v (p, p ; r) db v p =p surf (2.4) Fv (p) = [B v (T ) B v (T top )]t v (p, 0; r) + B v (T (p)) + t v (p, p; r) db v p =p where, taking benefit of the isotropic nature of the longwave radiation, the radiance L v of (2.3) has been replaced by the Planck function B v (T ) in units of flux, Wm 2 (here, and elsewhere, B v is assumed to always includes the π factor). T surf is the surface temperature, T 0+ that of the air just above the surface, T (p) is the temperature at pressure-level p, T top that at the top of the atmospheric model. The transmission t v is evaluated as the radiance transmission in a direction θ to the vertical such that r = sec θ is the diffusivity factor (Elsasser, 1942). Such an approximation for the integration over the angle is usual in radiative transfer calculations, and tests on the validity of this approximation have been presented by Rodgers and Walshaw (1966) and Liu and Schmetz (1988) among others. The use of the diffusivity factor gives cooling rates within 2% of those obtained with a 4-point Gaussian quadrature Vertical integration The integrals in (2.4) are evaluated numerically, after discretization over the vertical grid, considering the atmosphere as a pile of homogeneous layers. As the cooling rate is strongly dependent on local conditions p (2.2) (2.3) 8 IFS Documentation Cy31r1

3 of temperature and pressure, and energy is mainly exchanged with the layers adjacent to the level where fluxes are calculated, the contribution of the distant layers is simply computed using a trapezoidal rule integration, but the contribution of the adjacent layers is evaluated with a 2-point Gaussian quadrature, thus at the ith level pi 2 t v (p, p ; r) db v = db v (l)w l t v (p i, p l ; r) + 1 i 2 db v (j)[t v (p i, p j ; r) + t v (p i, p j 1 ; r)] (2.5) p =p surf 2 l=1 where p l is the pressure corresponding to the Gaussian root and w l is the Gaussian weight. db v (j) and db v (l) are the Planck function gradients calculated between two interfaces, and between mid-layer and interface, respectively Spectral integration The integration over wavenumber v is performed using a band emissivity method, as first discussed by Rodgers (1967). The longwave spectrum is divided into six spectral regions. (i) cm 1 and cm 1 (ii) cm 1 (iii) cm 1 and cm 1 (iv) cm 1 (v) cm 1 (vi) cm 1 and cm 1 corresponding to the centres of the rotation and vibration-rotation bands of H 2 O, the 15 µm band of CO 2, the atmospheric window, the 9.6 µm band of O 3, the 25 µm window region, and the wings of the vibration-rotation band of H 2 O, respectively. Over these spectral regions, band fluxes are evaluated with the help of band transmissivities precalculated from the narrow-band model of Morcrette and Fouquart (1985) See Appendix of Morcrette et al. (1986) for details. Integration of (2.4) over wavenumber ν within the kth spectral region gives the upward and downward fluxes as F + k (p) = {B k(t surf ) B k (T 0+ )}t Bk {ru(p surf, p), T U (p surf, p)} + B k (T p ) + p j=1 p =p surf t dbk {ru(p, p ), T U (p, p )} db k (2.6) F k (p) = {B k(t 0 ) B k (T )}t Bk {ru(p, 0), T U (p, 0)} B k (T p ) 0 p =p t dbk {ru(p, p), T U (p, p)} db k (2.7) The formulation accounts for the different temperature dependencies involved in atmospheric flux calculations, namely that on T p, the temperature at the level where fluxes are calculated, and that on T U, the temperature that governs the transmission through the temperature dependence of the intensity and half-widths of the lines absorbing in the concerned spectral region. The band transmissivities are nonisothermal accounting for the temperature dependence that arises from the wavenumber integration of the product of the monochromatic absorption and the Planck function. Two normalized band transmissivities are used for each absorber in a given spectral region: the first one for calculating the first right-handside term in (2.4), involving the boundaries; it corresponds to the weighted average of the transmission function by the Planck function t B (Up, T p, T U ) = v2 v 1 B v (T p )t v (Up, T U ) dv v2 v 1 B v (T p ) dv the second one for calculating the integral term in (2.4) is the weighted average of the transmission function by the derivative of the Planck function v2 v t db (Up, T p, T U ) = 1 {db(t p )/dt }t v (Up, T U ) dv v2 (2.9) v 1 {db(t p )/dt } dv IFS Documentation Cy31r1 9 (2.8)

4 Chapter 2: Radiation where Up is the pressure weighted amount of absorber. The effect on absorption of the Doppler broadening of the lines (important only for pressure lower than 10 hpa) is included simply using the pressure correction method of Fels (1979). A finite line width (assumed to represent the Doppler half-width of the line) is retained under low pressure conditions where the pure Lorentz line width (proportional to pressure) would normally become negligible (Giorgetta and Morcrette, 1995). In the scheme, the actual dependence on T p is carried out explicitly in the Planck functions integrated over the spectral regions. Although normalized relative to B(T p ) or db(t p )/dt, the transmissivities still depend on T U, both through Wien s displacement of the maximum of the Planck function with temperature and through the temperature dependence of the absorption coefficients. For computational efficiency, the transmissivities have been developed into Pade approximants t(up, T u ) = 2 i=0 c iu i/2 eff 2 j=0 d ju i/2 eff (2.10) where U eff = r(up)ψ(t U, Up) is an effective amount of absorber which incorporates the diffusivity factor r, the weighting of the absorber amount by pressure Up, and the temperature dependence of the absorption coefficients. The function Ψ(T U, Up) takes the form Ψ(T U, Up) = exp[a(up)(t U 250) + b(up)(t U 250) 2 ] (2.11) The temperature dependence due to Wien s law is incorporated although there is no explicit variation of the coefficients c i and d j with temperature. These coefficients have been computed for temperatures between and K with a 12.5 K step, and transmissivities corresponding to the reference temperature the closest to the pressure weighted temperature T U are actually used in the scheme The incorporation of the effects of clouds The incorporation of the effects of clouds on the longwave fluxes follows the treatment discussed by Washington and Williamson (1977). Whatever the state of the cloudiness of the atmosphere, the scheme starts by calculating the fluxes corresponding to a clear-sky atmosphere and stores the terms of the energy exchange between the different levels (the integrals in (2.4)) Let F 0 + (i) and F 0 (i) be the upward and downward clear-sky fluxes. For any cloud layer actually present in the atmosphere, the scheme then evaluates the fluxes assuming a unique overcast cloud of emissivity unity. Let F n + (i) and F n + (i) the upward and downward fluxes when such a cloud is present in the nth layer of the atmosphere. Downward fluxes above the cloud, and upward fluxes below the cloud, are assumed to be given by the clear-sky values F n + (i) = F 0 + (i) for i n Fn (i) = F0 (i) for i > n (2.12) Upward fluxes above the cloud (F n + (k) for k n + 1) and downward fluxes below it (Fn (k) for k > n) can be expressed with expressions similar to (2.5) provided the boundary terms are now replaced by terms corresponding to possible temperature discontinuities between the cloud and the surrounding air pk F n + (k) = {F + cld B(n + 1)}t(p k, p n+1 ; r) + B(k) + t(p k, p ; r) db p =p n 1 pn Fn (k) = {F cld B(n)}t(p k, p n ; r) + B(k) + t(p k, p ; r) db p =p k (2.13) where B(i) is now the total Planck function (integrated over the whole longwave spectrum) at level i, and F + cld and F cld are the longwave fluxes at the upper and lower boundaries of the cloud. Terms under the integrals correspond to exchange of energy between layers in clear-sky atmosphere and have already been computed in the first step of the calculations. This step is repeated for all cloudy layers. The fluxes for the actual atmosphere (with semi-transparent, fractional and/or multi-layered clouds) are derived from 10 IFS Documentation Cy31r1

5 a linear combination of the fluxes calculated in previous steps with some cloud overlap assumption in the case of clouds present in several layers. Let N be the index of the layer containing the highest cloud, C cld (i)) the fractional cloud cover in layer i, with C cld (0) = 1 for the upward flux at the surface, and with C cld (N + 1) = 1 and F N+1 = F 0 to have the right boundary condition for downward fluxes above the highest cloud. Whereas the maximum and random overlap assumptions are also available in the code (Morcrette and Fouquart, 1986), the maximum-random overlap assumption is operationally used in the ECMWF model, and the cloudy upward F + and downward F fluxes are obtained as F + (i) = F 0 + (i) for i = 1 i 2 F (i) = C cld (i 1)F + i 1 (i ) + i 1 C cld (n)f n + (i) {1 C cld (l)} for 2 i N + 1 N 1 F + (i) = C cld (N)F + N (i) + n=0 n=0 C cld (n)f + n (i) N l=n+1 l=n+1 {1 C cld (l)} for i N + 2 (2.14) In the case of semi-transparent clouds, the fractional cloudiness entering the calculations is an effective cloud cover equal to the product of the emissivity due to the condensed water and the gases in the layer by the horizontal coverage of the cloud layer, with the emissivity, ε cld, related to the condensed water amount by ε cld = 1 exp( k abs U LWP ) (2.15) where k abs is the condensed water mass absorption coefficient (in m 2 kg 1 ) following Smith and Shi (1992) The Rapid Radiation Transfer Model (RRTM) As stated in Mlawer et al. (1997), the objective in the development of RRTM has been to obtain an accuracy in the calculation of fluxes and heating rates consistent with the best line-by-line models. It utilizes the correlated-k method and shows its filiation to the Atmospheric and Environmental Research, Inc. (AER) line-by-line model (LBLRTM, Clough et al., 1989, 1992, Clough and Iacono, 1995) through its use of absorption coefficients for the relevant k-distributions derived from LBLRTM. Therefore the k-coefficients in RRTM include the effect of the CKD2.2 water vapour continuum (Clough et al., 1989). The main point in the correlated-k method (Lacis and Oinas, 1991; Fu and Liou, 1992) is the mapping of the absorption coefficient k(ν) from the spectral space (where it varies irregularly with wavenumber ν) to the g-space (where g(k) is the probability distribution function, i.e. the fraction of the absorption coefficients in the set smaller than k). The effect of this reordering is a rearrangement of the sequence of terms in the integral over wavenumber in the radiative transfer equation (RTE), which makes it equivalent to what would be done for monochromatic radiation. In the ECMWF model, no provision is presently taken for scattering in the longwave. Therefore, in order to get the downward radiance, the integration over the vertical dimension is simply done starting from the top of the atmosphere, going downward layer by layer. At the surface, the boundary condition (in terms of spectral emissivity, and potential reflection of downward radiance) is computed, then, in order to get the upward radiance, the integration over the vertical dimension is repeated, this time from the surface upward. The spectrally averaged radiance (between ν 1 and ν 2 ) emerging from an atmospheric layer is R = 1 ν1 { 1 } dν R 0 (ν) + [B(ν, T (t (ν 1 ν 2 ) ν)) R 0 (ν)] dt ν 2 t v (2.16) where R 0 is the incoming radiance to the layer, B(ν, T ) is the Planck function at wavenumber ν and temperature T, t ν is the transmittance for the layer optical path, and t ν is the transmittance at a point along the optical path in the layer. Under the mapping ν g, this becomes 1 { [ R = dg B eff (g, T g ) + [R 0 (g) B eff (g, T g )] exp k(g, P, T ) ρ z ]} (2.17) 0 cos φ IFS Documentation Cy31r1 11

6 Chapter 2: Radiation where B eff (g, T ) is an effective Planck function for the layer that varies with the layer s transmittance such as to ensure continuity of flux across layer boundaries for opaque conditions. The dependence of the transmittance is now written in terms of the absorption coefficient k(g, P, T ) at layer pressure P and temperature T, the absorber density ρ, the vertical thickness of the layer z, and the angle φ of the optical path. For a given spectral interval, the domain of the variable g is partitioned into subintervals (see Table 2.6, number of g-points), each corresponding to a limited range of k(g) values and for which a characteristic value κ j of the absorption coefficient is chosen. These κ j are then used to compute the outgoing radiance R = ( )] ρ z W j [B effj + (R 0j B effj ) exp κ j (2.18) cos φ j where W j is the size of the sub-intervals ( W j = 1). The accuracy of these absorption coefficients has been established by numerous and continuing highresolution validations of LBLRTM with spectroscopic measurements, in particular those from the Atmospheric Radiation Measurement program (ARM). Compared to the original RRTM (Mlawer et al., 1997), the version used at ECMWF has been slightly modified to account for cloud optical properties and surface emissivity defined for each of the 16 bands over which spectral fluxes are computed. For efficiency reason, the original number of g-points (256 = 16 16) has been reduced to 140 (see Table 2.6). Other changes are the use of a diffusivity approximation (instead of the three-angle integration over the zenith angle used in the original scheme) to derive upward and downward fluxes from the radiances, and the modification of the original cloud random overlapping assumption to include (to the same degree of approximation as used in the operational SW scheme) a maximum-random overlapping of cloud layers. Given the monochromatic form of the RTE, the vertical integration is simply carried out one layer at a time from the top-of-the-atmosphere to the surface to get the downward fluxes. The downward fluxes at the surface are then used with the spectral surface emissivities and the surface temperature to get the upward longwave fluxes in each of the 140 subintervals. Then the upward fluxes are obtained in a similar fashion from the surface to the ToA. For the relevant spectral intervals of the RRTM schemes, ice cloud optical properties are derived from Ebert and Curry (1992), and water cloud optical properties from Fouquart (1987). Whereas in the previous operational scheme the cloud emissivity used to compute the effective cloud cover is defined over the whole LW spectrum from spectrally averaged mass absorption coefficients and the relevant cloud water and/or ice paths (following Smith and Shi, 1992), in RRTM, the cloud optical thickness is defined as a function of spectrally varying mass absorption coefficients and relevant cloud water and ice paths, and is used within the true cloudy fraction of the layer. Alternate sets of cloud optical properties are also available for RRTM, based on Savijarvi and Raisanen (1997) for liquid water clouds, and Fu et al. (1998) for ice clouds. 2.3 SHORTWAVE RADIATION The rate of atmospheric heating by absorption and scattering of shortwave radiation is T t = g F SW c p p (2.19) where F SW is the net total shortwave flux (the subscript SW will be omitted in the remainder of this section). [ 2π { +1 }] F(δ) = dν dφ µl ν (δ, µ, φ) dµ (2.20) is the diffuse radiance at wavenumber ν, in a direction given by the azimuth angle, φ, and the zenith angle, θ, with µ = cos θ. In (2.20), we assume a plane parallel atmosphere, and the vertical coordinate is the optical depth δ, a convenient variable when the energy source is outside the medium 0 δ(p) = βv ext (p ) dp (2.21) p 12 IFS Documentation Cy31r1

7 βν ext (p) is the extinction coefficient, equal to the sum of the scattering coefficient βν sca of the aerosol (or cloud particle absorption coefficient βν abs ) and the purely molecular absorption coefficient k ν. The diffuse radiance L ν is governed by the radiation transfer equation µ dl ν(δ, µ, φ) dδ = L ν (δ, µ, φ) ϖ ν(δ) P ν (δ, µ, φ, µ 0, φ 0 )Eν 0 exp( δ/µ r ) 4 ϖ 2π { +1 } ν(δ) dφ Φ ν (δ, µ, φ, µ, φ )L ν (δ, µ, φ ) dµ (2.22) Eν 0 is the incident solar irradiance in the direction µ 0 = cos θ 0, ϖ ν, is the single scattering albedo (= βν sca /k v ) and Φ(δ, µ, φ, µ, φ ) is the scattering phase function which defines the probability that radiation coming from direction (µ, φ ) is scattered in direction (µ, φ). The shortwave part of the scheme, originally developed by Fouquart and Bonnel (1980) solves the radiation transfer equation and integrates the fluxes over the whole shortwave spectrum between 0.2 and 4 µm. Upward and downward fluxes are obtained from the reflectances and transmittances of the layers, and the photon-path-distribution method allows to separate the parametrization of the scattering processes from that of the molecular absorption Spectral integration Solar radiation is attenuated by absorbing gases, mainly water vapour, uniformly mixed gases (oxygen, carbon dioxide, methane, nitrous oxide) and ozone, and scattered by molecules (Rayleigh scattering), aerosols and cloud particles. Since scattering and molecular absorption occur simultaneously, the exact amount of absorber along the photon path length is unknown, and band models of the transmission function cannot be used directly as in longwave radiation transfer (see Section 2.2). The approach of the photon path distribution method is to calculate the probability Π(U) du that a photon contributing to the flux F cons in the conservative case (i.e., no absorption, ω ν = 1, k ν = 0) has encountered an absorber amount between U and U + du. With this distribution, the radiative flux at wavenumber v is related to F cons by F v = F cons Π(U) exp( k ν U) du (2.23) 0 and the flux averaged over the spectral interval ν can then be calculated with the help of any band model of the transmission function t ν F = 1 F ν dν = F cons Π(U)t ν (U) dν (2.24) ν ν To find the distribution function Π(U), the scattering problem is solved first, by any method, for a set of arbitrarily fixed absorption coefficients k 1, thus giving a set of simulated fluxes F k1. An inverse Laplace transform is then performed on (2.23) (Fouquart, 1974). The main advantage of the method is that the actual distribution Π(U) is smooth enough that (2.23) gives accurate results even if Π(U) itself is not known accurately. In fact, Π(U) needs not be calculated explicitly as the spectrally integrated fluxes are 0 F = F cons t v ( U ) F = F cons t v ( U 1/2 ) in the limiting case of weak absorption in the limiting case of strong absorption where U = 0 Π(U)U du and U 1/2 = 0 Π(U)U 1/2 du. The atmospheric absorption in the water vapour bands is generally strong, and the scheme determines an effective absorber amount U e between U and U 1/2 derived from U e = ln(f ke /F cons )/k e (2.25) where k e is an absorption coefficient chosen to approximate the spectrally averaged transmission of the clear sky atmosphere 1 k e = ln(t v (U tot /µ 0 )) (2.26) U tot /µ 0 where U tot is the total amount of absorber in a vertical column and µ 0 = cos θ 0. Once the effective absorber amounts of H 2 O and uniformly mixed gases are found, the transmission functions are computed IFS Documentation Cy31r1 13

8 Chapter 2: Radiation using Pade approximants t ν (U) = N i=0 a iu i 1 N j=0 b ju j 1 (2.27) Absorption by ozone is also taken into account, but since ozone is located at low pressure levels for which molecular scattering is small and Mie scattering is negligible, interactions between scattering processes and ozone absorption are neglected. Transmission through ozone is computed using (2.24) where U O3 the amount of ozone is U d O 3 = M U u O 3 = r 0 du O3 p 0 for the downward transmission of the direct solar beam p s du O3 + U d O 3 (p surf ) for the upward transmission of the diffuse radiation r = 1.66 is the diffusivity factor (see Section 2.2), and M is the magnification factor (Rodgers, 1967) used instead of r to account for the sphericity of the atmosphere at very small solar elevations M = 35/ µ (2.28) To perform the spectral integration, it is convenient to discretize the solar spectrum into subintervals in which the surface reflectance, molecular absorption characteristics, and cloud optical properties can be considered as constants. One of the main causes for such a spectral variation is the sharp increase in the reflectivity of the vegetation in the near-infrared. Also, water vapour does not absorb below 0.69 µm nor do liquid water clouds. Till June 2000, the ECMWF shortwave scheme considered only two spectral intervals, one for the visible ( µm), one for the near-infrared ( µm) parts of the solar spectrum. From June 2000 to April 2002, the near-infrared interval was sub-divided into three intervals ( µm) to account better for the spectral variations of the cloud optical properties. Till April 2002, all the molecular absorption coefficients (for O 3, H 2 O, uniformly mixed gases) were derived from statistical models of the transmission function using spectroscopic parameters derived from various versions of the HITRAN database (Rothman et al., 1986, 1992). In April 2002, following the recomputation of all the molecular absorption coefficients from an updated version of the shortwave lineby-line model of Dubuisson et al. (1996) using spectroscopic data from HAWKS (2000), the ultraviolet and visible part of the spectrum are now considered in three spectral intervals ( µm) making the scheme having a total of six spectral intervals over which the aerosol and cloud optical properties are also defined. The cut-off at 0.69 µm allows the scheme to be more computational efficient, in as much as the interactions between gaseous absorption (by water vapour and uniformly mixed gases) and scattering processes are accounted for only in the near-infrared interval(s) Vertical integration Considering an atmosphere where a fraction Ccld tot (as seen from the surface or the top of the atmosphere) is covered by clouds (the fraction Ccld tot depends on which cloud-overlap assumption is assumed for the calculations), the final fluxes are given as a weighted average of the fluxes in the clear sky and in the cloudy fractions of the column F (j) = CcldF tot cld (j) + (1 Ctot cld)f clr where the subscripts clr and cld refer to the clear-sky and cloudy fractions of the layer, respectively. In contrast to the scheme of Geleyn and Hollingsworth (1979), the fluxes are not obtained through the solution of a system of linear equations in a matrix form. Rather, assuming an atmosphere divided into homogeneous layers, the upward and downward fluxes at a given layer interface j are given by F (j) = F 0 N k=j T bot (k) F + (j) = F (j)r top (j 1) (2.29) 14 IFS Documentation Cy31r1

9 where R top (j) and T bot (j) are the reflectance at the top and the transmittance at the bottom of the jth layer. Computation of the values of R top starts at the surface and works upwards, whereas determining values of T bot starts at the top of the atmosphere and works downward. R top and T bot account for the presence of cloud in the layer by using R top = C cld R cld + (1 C cld )R clr T bot = C cld T cld + (1 C cld )T clr (2.30) where C cld is the cloud fractional coverage of the layer within the cloudy fraction C tot cld (a) Cloudy fraction layer of the column. R tcdy and R bcdy are the reflectance at the top and transmittance at the bottom of the cloudy fraction of the layer calculated with the Delta-Eddington approximation. Given δ c, δ a, and δ g, the optical thicknesses for the cloud, the aerosol and the molecular absorption of the gases (= k e U), respectively, and g c and g a the cloud and aerosol asymmetry factors, R tcdy and R bcdy are calculated as functions of the total optical thickness of the layer δ = δ c + δ a + δ g (2.31) of the total single scattering albedo of the total asymmetry factor g = ϖ = δ c + δ a δ c + δ a + δ g (2.32) δ c g c + δ a g a (2.33) δ c + δ a δ c + δ a of the reflectance R of the underlying medium (surface or layers below the jth interface), and of the cosine of an effective solar zenith angle µ eff (j) which accounts for the decrease of the direct solar beam and the corresponding increase of the diffuse part of the downward radiation by the upper scattering layers µ eff (j) = [(1 Ccld(j))/µ eff + rccld(j)] eff 1 (2.34) with Ccld eff (j) the effective total cloudiness over level j and C eff cld(j) = 1 N i=j+1 [ E(i) = 1 exp (1 ϖ c(i)g c (i) 2 ] )δ c (i) µ (1 C cld (i)e(i)) (2.35) (2.36) δ c (i), ϖ c (i) and g c (i) are the optical thickness, single scattering albedo and asymmetry factor of the cloud in the ith layer, and r is the diffusivity factor. The scheme follows the Eddington approximation first proposed by Shettle and Weinman (1970), then modified by Joseph et al. (1976) to account more accurately for the large fraction of radiation directly transmitted in the forward scattering peak in case of highly asymmetric phase functions. Eddington s approximation assumes that, in a scattering medium of optical thickness δ, of single scattering albedo ω, and of asymmetry factor g, the radiance L entering (2.17) can be written as L(δ, µ) = L 0 (δ) + µl 1 (δ) (2.37) In that case, when the phase function is expanded as a series of associated Legendre functions, all terms of order greater than one vanish when (2.20) is integrated over µ and φ. The phase function is therefore given by P (Θ) = 1 + β 1 (Θ)µ where Θ is the angle between incident and scattered radiances. The integral in (2.20) thus becomes 2π 0 { +1 } dφ p(µ, φ, µ, φ )L(µ, φ ) dµ = 4π(L 0 + πl 1 ) (2.38) 1 IFS Documentation Cy31r1 15

10 Chapter 2: Radiation where is the asymmetry factor. g = β 1 3 = 1 +1 P (Θ)µ dµ 2 1 Using (2.38) in (2.20) after integrating over µ and dividing by 2π, we get µ d dδ (L 0 + µl 1 ) = (L 0 + µl 1 ) + ϖ(l 0 + gµl 1 ) + 1/4ϖF 0 exp( δ/µ 0 )(1 + 3gµ 0 µ) (2.39) We obtain a pair of equations for L 0 and L 1 by integrating (2.39) over µ dl 0 dδ = 3(1 ϖ)l ϖf 0 exp( δ/µ 0 ) dl 1 dδ = (1 ϖg)l ϖgµ 0F 0 exp( δ/µ 0 ) (2.40) For the cloudy layer assumed non-conservative (ϖ < 1), the solutions to (2.39) and (2.40), for 0 δ δ, are L 0 (δ) = C 1 exp( Kδ) + C 2 exp(+kδ) α exp( δ/µ 0 ) L 1 (δ) = P {C 1 exp( Kδ) C 2 exp(+kδ) β exp( δ/µ 0 )} (2.41) where K = {3(1 ϖ)(1 ϖg)} 1/2 P = {3(1 ϖ)/(1 ϖg)} 1/2 α = 3ϖF 0 µ 0 {1 + 3g(1 ϖ)}/{4(1 K 2 µ 2 0)} β = 3ϖF 0 µ 0 {1 + 3g(1 ϖ)µ 2 0}/(4(1 K 2 µ 2 0)) The two boundary conditions allow to solve the system for C 1 and C 2 ; the downward directed diffuse flux at the top of the atmosphere is zero, that is [ F (0) = L 0 (0) + 2 ] 3 L 1(0) = 0 which translates into (1 + 2P/3)C 1 + (1 2P/3)C 2 = α + 2β/3 (2.42) The upward directed flux at the bottom of the layer is equal to the product of the downward directed diffuse and direct fluxes and the corresponding diffuse and direct reflectance (R d and R, respectively) of the underlying medium { F + (δ ) = L 0 (δ ) 2 } 3 L 1(δ ) = R {L 0 (δ ) + 2 } 3 L 1(δ ) + R d µ 0 F 0 exp( δ /µ 0 ) which translates into {1 R 2(1 + R )P/3}C 1 exp( Kδ ) + {1 R + 2(1 + R )P/3}C 2 (+Kδ ) = {(1 R )α 2(1 + R )β/3 + R d µ 0 F 0 } exp( δ /µ 0 ) (2.43) In the Delta-Eddington approximation, the phase function is approximated by a Dirac delta function forward-scatter peak and a two-term expansion of the phase function P (θ) = 2f(1 µ) + (1 f)(1 + 3g µ) 16 IFS Documentation Cy31r1

11 where f is the fractional scattering into the forward peak and g the asymmetry factor of the truncated phase function. As shown by Joseph et al. (1976), these parameters are f = g 2 g = g/(g + 1) (2.44) The solution of the Eddington s equations remains the same provided that the total optical thickness, single scattering albedo and asymmetry factor entering (2.39) and (2.43) take their transformed values δ = (1 + ϖf)δ ω = (1 f)ϖ 1 ϖf (2.45) Practically, the optical thickness, single scattering albedo, asymmetry factor and solar zenith angle entering (2.39) (2.43) are δ, ϖ, g and µ eff defined in (2.33) and (2.34). (b) Clear-sky fraction of the layers In the clear-sky part of the atmosphere, the shortwave scheme accounts for scattering and absorption by molecules and aerosols. The following calculations are practically done twice, once for the clear-sky fraction (1 Ccld tot) of the atmospheric column µ with equal to µ 0, simply modified for the effect of Rayleigh and aerosol scattering, the second time for the clear-sky fraction of each individual layer within the fraction Ccld tot of the atmospheric column containing clouds, with µ equal to µ e. As the optical thickness for both Rayleigh and aerosol scattering is small, R clr (j 1) and T clr (j), the reflectance at the top and transmittance at the bottom of the jth layer can be calculated using respectively a first- and a second-order expansion of the analytical solutions of the two-stream equations similar to that of Coakley Jr. and Chylek (1975). For Rayleigh scattering, the optical thickness, single scattering albedo and asymmetry factor are respectively δ R, ϖ R = 1 and g R = 0, so that δ R R R = 2µ + δ R 2µ T R = (2µ + δ R ) (2.46) The optical thickness δ R of an atmospheric layer is simply δ R = δ {p(j) p(j 1)}/p surf (2.47) where δr is the Rayleigh optical thickness of the whole atmosphere parametrized as a function of the solar zenith angle (Deschamps et al., 1983) For aerosol scattering and absorption, the optical thickness, single scattering albedo and asymmetry factor are respectively δ a, ϖ a, with 1 ϖ a 1 and g a, so that den = 1 + {1 ϖ a + back(µ e )ϖ a }(δ a /µ e ) + (1 ϖ a ){1 ϖ a + 2 back(µ e )ϖ a }(δ 2 a/µ 2 e) (2.48) R(µ e ) = (back(µ e)ϖ a δ a )/µ a den T (µ e ) = 1/den (2.49) where back(µ e ) = (2 3µ e g a )/4 is the backscattering factor. Practically, R clr and T clr are computed using (2.49) and the combined effect of aerosol and Rayleigh scattering comes from using modified parameters corresponding to the addition of the two scatterers with provision for the highly asymmetric aerosol phase function through Delta-approximation of the IFS Documentation Cy31r1 17

12 Chapter 2: Radiation forward scattering peak (as in (2.40) and (2.41)). δ + = δ R + δ a (1 ϖ a g 2 a) g + = ϖ + = g a 1 + g a δ a (δ R + δ a ) δ R ϖ R + δ a ϖ a (1 ga) 2 δ a + δ a δ R + δ a 1 ϖ a ga 2 (2.50) As for their cloudy counterparts, R clr and Tċlr must account for the multiple reflections due to the layers underneath R clr = R(µ e ) + R T (µ e )(1 R R ) (2.51) and R is the reflectance of the underlying medium R = R t (j 1) and r is the diffusivity factor. Since interactions between molecular absorption and Rayleigh and aerosol scattering are negligible, the radiative fluxes in a clear-sky atmosphere are simply those calculated from (2.27) and (2.45) attenuated by the gaseous transmissions (2.25) Multiple reflections between layers To deal properly with the multiple reflections between the surface and the cloud layers, it should be necessary to separate the contribution of each individual reflecting surface to the layer reflectance and transmittances in as much as each such surface gives rise to a particular distribution of absorber amount. In the case of an atmosphere including N cloud layers, the reflected light above the highest cloud consists of photons directly reflected by the highest cloud without interaction with the underlying atmosphere, and of photons that have passed through this cloud layer and undergone at least one reflection on the underlying atmosphere. In fact, (2.22) should be written F = N F cl P 1 (U)t v (U)dv (2.52) i=0 0 where F cl and P 1 (U) are the conservative fluxes and the distributions of absorber amount corresponding to the different reflecting surfaces. Fouquart and Bonnel (1980) have shown that a very good approximation to this problem is obtained by evaluating the reflectance and transmittance of each layer (using (2.39) and (2.45)) assuming successively a non-reflecting underlying medium (R = 0), then a reflecting underlying medium (R 0). First calculations provide the contribution to reflectance and transmittance of those photons interacting only with the layer into consideration, whereas the second ones give the contribution of the photons with interactions also outside the layer itself. From those two sets of layer reflectance and transmittances (T t0, T b0 ) and (R t, T b ) respectively, effective absorber amounts to be applied to computing the transmission functions for upward and downward fluxes are then derived using (2.23) and starting from the surface and working the formulas upward U e0 = ln (T b0/t bc )/k e U e = ln (T b /T bc )/k e U + e0 = ln (R t0/r tc )/k e (2.53) U + e = ln (R t /R tc )/k e where R tc and T bc are the layer reflectance and transmittance corresponding to a conservative scattering medium. Finally the upward and downward fluxes are obtained as F + (j) = F 0 {R t0 t ν (U e0 + ) + (R t R t0 )t ν (U + e )} (2.54) F (j) = F 0 {T b0 t ν (U e0 + ) + (T b T b0 )t ν (U e )} (2.55) 18 IFS Documentation Cy31r1

13 2.3.4 Cloud shortwave optical properties As seen in Subsection 2.3.2(a), the cloud radiative properties depend on three different parameters: the optical thickness δ c, the asymmetry factor g c, and the single scattering albedo ϖ c. Presently the cloud optical properties are derived from Fouquart (1987) for the water clouds, and Ebert and Curry (1992) for the ice clouds. The optical thickness δ c is related to the cloud liquid water amount U LWP by δ c = 3U LWP 2r e where r e is the mean effective radius of the size distribution of the cloud water droplets. Presently r e is parametrized as a linear function of height from 10 µm at the surface to 45 µm at the top of the atmosphere, in an empirical attempt at dealing with the variation of water cloud type with height. Smaller water droplets are observed in low-level stratiform clouds whereas larger droplets are found in mid-level cumuliform water clouds. In the two-, four-, and six-spectral interval versions of the shortwave radiation scheme, the optical properties of liquid water clouds are defined from Fouquart (1987) and those for ice clouds from Ebert and Curry (1992). Alternative optical properties are also available for liquid water clouds (Slingo, 1989) and ice clouds (Fu, 1996). The effective radius of the liquid water cloud particles is computed from the cloud liquid water content using the diagnostic formulation of Martin et al. (1994) and specified concentrations of cloud concentration nuclei over land and ocean. For ice clouds, the effective dimension of the cloud particles is diagnosed from temperature using a revision of the formulation by Ou and Liou (1995). 2.4 HORIZONTAL INTERPOLATION As stated in the introduction, the cost of the radiation scheme described in the previous sections is prohibitive if it were used to compute the radiative fluxes at every time step and every grid point of the model. In order to cut down the computing costs, the full radiation scheme is only called every 3 hours (every 1 hour during the first 12 hours used for data assimilation) (the so-called full radiation time steps) and on a reduced grid interpolated from the full physical grid. A spatial and temporal interpolation thus provides the relevant interaction of the shortwave radiative fluxes with the solar zenith angle at every time step and every grid point Temporal interpolation To do so, a shortwave transmissivity is defined at each model level such that F s = τ e S 0 (2.56) where F s is the net solar (shortwave) flux and S 0 is the solar flux at the top of the atmosphere. F s is defined only for a full radiation time step. At every time step, the net solar fluxes are computed therefore from the transmissivity derived for the last full radiation time step, using (2.56) with the correct solar angle for every grid point. The net longwave fluxes at kept at the values given by the full radiation calculation Spatial interpolation Full radiation computations are now performed using the so-called halo configuration that can be defined according to needs for the various spatial resolutions. The previous spatial sampling (operational till Cy26r1), was done only in the longitudinal direction. It was going from one out of four points prevalent in sub-tropical and tropical latitudes and reduced gradually to every point in polar areas. On output, Lagrangian cubic interpolation was used. The scheme IFS Documentation Cy31r1 19

14 Chapter 2: Radiation Table 2.1 Possible resolutions of the new interpolation scheme for radiation computations for the various dynamical resolutions of the ECMWF forecast system. Res NDLON RadRes N/A N/A NDLON is the maximum number of longitude points for the reference configuration with radiative (and other physics) computations at all grid points, 2, 3 and 4 correspond to a larger grid for radiative computations. Default values for model configurations from Cy26r3 are in bold. Note that default T95 does not use a larger grid for radiation. The maximum number of longitude points for the radiative computations can be obtained from the equivalent value of Res. A maximum of 42 and 128 longitude points is respectively used for radiative computations for RadRes = 21 and 63. Table 2.2 Speed-up factor of the various radiation configurations relative to a computation at all grid points (configuration 1), for different horizontal resolutions. Res N/A N/A is the previous operational configuration with sampling up to one point out of four in each latitude band, 2, 3 and 4 correspond to the relevant resolution in Table 2.1. Default values for model configurations from Cy26r3 are in bold. worked efficiently on vector systems with less than 100 processors and scalar systems with about 1000 processors. The only real problem was the complexity of the message passing, a direct result of the use of a non-standard grid for radiation calculations. The new interface for radiation computations was developed to address this complexity, and uses a standard IFS model grid, but with a coarser resolution than the current model grid. Further, interpolation between model and radiation grids are performed using the interfaces already existing within the IFS for the semi-lagrangian interpolation, and as a result should reduce future code maintenance. By using such a standard grid for radiation computations, there is no longer a load balance issue, as each processor is given an equal number of grid points for model and radiation grids. A new grid is computed, independent of that for the rest of the physics, over which input fields are averaged using the standard interpolation routines. Then radiation computations are done, and output fluxes are interpolated back to the reduced grid, at times of full radiation computations. This new halorelated grid can be chosen differently with the forecast application (seasonal runs, EPS, high-resolution 10-day forecasts). Table 2.1 presents the various basic model resolutions together with the resolution made available, by default, for radiation computations by the new interface, whereas Table 2.2 presents the speed-up factor introduced by the various radiation configurations corresponding to horizontal resolutions used for different applications. 20 IFS Documentation Cy31r1

15 2.5 INPUT TO THE RADIATION SCHEME Model variables Temperature values are needed at the boundaries of the layers, where the fluxes are computed. They are derived from the full level temperatures with a pressure weighted interpolation p k (p k+1 p k+1/2 ) T k+1/2 = T k p k+1/2 (p k+1 p k ) + T p k+1/2 (p k+1/2 p k ) k+1 p k+1/2 (p k+1 p k ) (2.57) At the bottom of the atmosphere, either the surface temperature or the temperature at 2 m is used, while at the top of the atmosphere the temperature is extrapolated from the first full level and second half level temperatures Clouds Cloud fraction, and liquid/ice water content is provided in all layers by the cloud scheme Aerosols The aerosol climatology used in the operational model up to Cy26r1 was given as annual mean geographical distributions defined from T5 spectral coefficients, for different aerosol types, respectively, maritime, continental, urban and desert, plus a uniformly distributed stratospheric background aerosols, with fixed vertical distributions, following Tanre et al. (1984). In the last fifteen years, chemical and/or transport models have addressed the life cycles of various aerosol types and attempted an inventory of their spatio-temporal distributions. Out of these studies, a new climatology for the annual cycle of the aerosol distribution of various aerosol types has been compiled by Tegen et al. (1997), which has been implemented in the ECMWF forecast system from Cy26r3 onwards. Table 2.3 describes the characteristics of the aerosol components for each tropospheric aerosol type and Table 2.4 compares the maximum optical thicknesses in the old and new climatologies Carbon dioxide, ozone and trace gases Carbon dioxide, methane, nitrous oxide, CFC-11 and CFC-12 have constant volume concentrations of 353 ppm, 1.72 ppm, 0.31 ppm, 280 ppt, and 484 ppt, respectively (IPCC/SACC, 1990), except in ERA-40 for the variation in concentrations is derived from (IPCC/SACC, 1995). Two climatologies are available for the ozone distribution. In the first one (NOZOCL = 0), the ozone mixing ratio q O3 depends on height, latitude, longitude and season. Its vertical distribution is assumed to be such that its integral from 0 to the pressure p is p 0 q O3 dp = a 1 + (b/p) 3/2 (2.58) The constants a and b are related to the total amount of ozone and the height of its maximum mixing ratio. They are imposed in terms of a limited series of spherical harmonics (T10) for the geographical distribution and a Fourier series for the seasonal variation. The total amount of ozone was taken from London et al. (1976) and the altitude of the maximum concentration was derived from Wilcox and Belmont (1977). Plots of these values can be found in the Appendix. In the second climatology (NOZOCL = 1), the ozone mixing ratio q O3 depends on height, latitude and month, and is taken from Fortuin and Langematz (1994) Ground albedo and emissivity The background land albedo, α sb, is interpolated to the model grid from the monthly mean values of a snow-free albedo produced for the combined years. The albedo for that dataset was computed using the method of Sellers et al. (1996), but with new maps of soil reflectance, new values of vegetation reflectance and the biophysical parameters described in Los et al. (2000). More information on the original data and plots of the monthly mean albedo are shown in Chapter 10. Spectral albedos for parallel and diffuse radiation are needed by the radiative code. In addition, the surface energy balance equation (see Chapter 3 on vertical diffusion) needs a spectrally integrated IFS Documentation Cy31r1 21

16 Chapter 2: Radiation Table 2.3 Characteristics of the aerosol components for each tropospheric aerosol type in the new climatology for cycle CY26R3 of the ECMWF model (adapted from Hess et al. (1998)). Number Volume Mass Density Type RH(%) Component (cm 1 ) (µm 3 /m 3 ) (µg 3 /m 3 ) (g/cm 3 ) Continental 80 organic Maritime 95 sulphate Desert 50 dust-like Urban 80 black carbon Insoluble 4.00E E E Water soluble 7.00E E E soot 8.30E E E Water soluble 1.50E E E sea salt (accum.) 2.00E E E sea salt (coarse) 3.20E E E Water soluble 2.00E E E Mineral (nuclei) 2.70E E E Mineral (accum.) 3.05E E E Mineral (coarse) 1.42E E E Insoluble 1.50E E E Water soluble 2.80E E E Soot 1.30E E E Type: First definition (e.g. continental) is the aerosol component as known within both the ECMWF model and the OPAC software; second definition (e.g. organic) is the 3D distribution to which it is linked in the climatology of Tegen et al. (1997). RH is the relative assumed for the computations of the relevant optical properties. The nuclei, accumulation, and coarse modes refer to various size ranges for the component particles. Table 2.4 Maximum optical thickness in the two aerosol climatologies. OLD Annual January July NEW Continental Organic Maritime Sulphate Desert Dust-like Urban Black carbon Background trop Background stratos Background stratos. Aerosol types of the new and old climatologies are paired according to the dominant components in each mix. 22 IFS Documentation Cy31r1

17 Table 2.5 Diffuse and parallel albedo and window emissivity for each tile. Tile Open Sea Interception Low Exposed High Shaded Bare Description sea ice layer vegetation snow vegetation snow ground Diffuse 0.06 Ebert and α sb α sb α sn α sb 0.15 α sb albedo Curry (1993) Parallel Taylor Ebert and α sb α sb α sn α sb 0.15 α sb albedo et al. (1996) Curry (1993) Window emissivity parallel+diffused albedo, specified for each independent surface functional unit, tile. The procedure is summarized in Table 2.5. Over open water, the surface albedo for direct parallel radiation is a fit to low-flying aircraft measurements over the ocean given by Taylor et al. (1996) α sp = 1.1µ 1.4 (2.59) For sea ice, monthly values based on Ebert and Curry (1993) albedos for the Arctic Ocean are interpolated to the forecast time. The bare sea ice albedo value in Ebert and Curry is taken as a representative value for summer, and the dry snow albedo value is used for the winter months. Values for the Antarctic are shifted by six months. Separate values for visible and near-infrared spectral bands are used. The timevarying snow albedo (α sn, see Chapter 7), is used for the exposed snow tile only. Finally, the average of the diffuse and parallel albedos are spectrally integrated for each tile. The thermal emissivity of the surface outside the cm 1 spectral region is assumed to be 0.99 everywhere. In the window region, the spectral emissivity is constant for open water, sea ice, the interception layer and exposed snow tiles. For low and high vegetation and for shaded snow the emissivity depends on the water content in the top soil layer. Emissivity decreases linearly from 0.96 for soils at or above field capacity to 0.93 for soils at or below permanent wilting point. The same formulation is used for bare ground, except for desert areas (α sb > 0.3), where a value of 0.93 is used independently of the soil water content. Finally, a broadband emissivity is obtained by convolution of the spectral emissivity and the Planck function at the skin temperature Solar zenith angle Equations to compute the annual variation of the solar constant I, the solar declination δ s and the difference between solar time and official time can be found in Paltridge and Platt (1976). These equations are used to give the cosine of the solar angle at the ground. Because of the curvature of the earth, the zenith angle is not quite constant along the path of a sun ray. Hence the correction applied to µ a 0 to give an average µ 0 for the atmosphere is µ 0 = (µ a 0 )2 + H a H a ( 2 + H a ) (µ a 0 )2 (2.60) where a is the earth radius and H is the atmospheric equivalent height. H/a is fixed at THE RADIATION CODE Routine RADHEAT or RADHEATN (depending whether the diagnostic or prognostic cloud scheme is used) is called at every time step to compute the radiative fluxes and heating using the solar zenith angle computed in CPGLAG and emissivities and transmissivities (PEMTU, PTRSOL) computed at full radiation time steps in RADINT. or RADINTG (see Subsection 2.6.2). The other routines are called either once at the beginning of the run (SUECRAD and below) or once per full radiation step at the first row (ECRADFR and below), or at every full radiation time step for all rows. In this section, we briefly describe the function of each routine. IFS Documentation Cy31r1 23